Field
The present disclosure relates generally to automated systems and methods for student instruction and teacher training and, more specifically but not exclusively, to systems and methods using mathematical reasoning blocks.
Description of the Related Art
This section introduces aspects that may help facilitate a better understanding of the embodiments disclosed herein. Accordingly, the statements of this section are to be read in this light and are not to be understood as admissions about what is in the prior art or what is not in the prior art.
American elementary school math students score among the top children from industrialized countries on tests designed to measure mathematical knowledge. Yet, by the end of high school, they find themselves among the lowest performers on such standardized tests. The American educational system needs a remedy for this situation so that at-risk students can find success in pre-algebra and algebra courses—the gateway subjects to college and the foundation of the technical expertise needed to solve many of the world's most pressing problems.
An inability to balance intuitive, whole-language approaches to teaching mathematics with more-traditional, analytic approaches is, in many ways, at the root of the entire problem. Whether because of student laziness, poor teacher pedagogy, technological limitations or the failure of imagination, mathematics often ends up being taught and grasped in large, memorized chunks. The problem becomes that students who memorize math problem solutions like phrases in a phrase book often find themselves lacking a sense of the individual parts making up the solutions as wholes. They fail to acquire knowledge of how to rearrange these smaller constituent parts to express new ideas or solve new problems.
Most computerized instruction now takes place in one of several modes, each characterized by some form of this imbalance.
In one mode, instruction is highly traditional. It occurs at the surface of the subject, resulting in multiple-choice environments that are overly mechanical and prone to intuitive guess-work on the part of students—a complete overemphasis on intuitive wholes.
In another traditional mode, programs have been designed that try to ensure an understanding of the steps involved in solving a math problem. But current attempts at success with this mode have resulted in rigid, unalterable templates that eliminate flexibility in the solution of problems.
Graphing calculators or geometric-sketchpad type environments—with their holistic, global linking of functional table-graph-algebraic representations—provide the most ubiquitous and modern example of the imbalance that results from this overemphasis on systems of wholes. Students may manipulate whole representations—entire graphs or tables or equations. But it is the device or program that automates the computations in and among representations, taking much of the actual analytic work out of users' hands so they can focus on the meaning behind a given real life problem or situation.
This functional table-figure/graph-algebraic approach to teaching mathematics is supported by a body of academic work that spans the mid-1980s to the present. According to Kieran (2007, p. 709), Fey and Good (1985) formulated the stance that dominates mathematics pedagogy in the U.S. today. Their idea was: “that practicing manipulative skills should be replaced with the study of families of elementary functions, a shift that ‘places the function concept at the heart of the curriculum’ (p. 49).” The emphasis is on real world situations that have relationships that can be described and modeled in functional terms.
Not everyone agrees with this approach. Exemplifying the opposition's stance is Pimm (1995). He writes, “There is currently a rapid process of redefinition of algebra, triggered I feel more by the potentialities of these new [digital] systems and the drawbacks of an over-fragmented mathematics curriculum than by any novel epistemological insight.” (p. 104). Kieran summed up the situation that has resulted in U.S. educational circles today, writing, “The orientation towards the solving of realistic problems, with the aid of technological tools, allows for an algebraic content that is less manipulation oriented” (p. 710). Students are able to concentrate on the meaning of what is being done, but computers and calculators are doing most of the doing.
In Teaching Math as a Language (2007) and in seminars before and after the publication of that book, the inventor agreed with this stance, arguing that like a language, mathematics has both structure and meaning and that the math educational community needed to find a balance between the abstract syntactical, grammatical, analytical based concerns of traditional mathematics and the realistic functional, meaning-based, holistic concerns of the now-dominant reform-math approaches.
Whether a person embraces the function-based realism that has prevailed in U.S. math pedagogy for the last twenty years, the purported high-abstraction of traditional approaches that is experiencing a small resurgence or a balance between both schools, in the end the real question boils down to one of scalability. For despite the widespread acceptance of reform-math approaches, no one has been able to scale theory up into widespread practice in a way that produces measureable results. As the U.S. looks towards the coming decades of the twenty-first century, there remains this final and most important of all math educational needs, a solution to improving math education that is easily scalable. Thus, new and improved systems and/or methods that provide a math pedagogy that is not just theoretically sound but strongly scalable need to be developed to answer these needs.
References that generally discuss these areas of education include the following.    Fey, J., & Good, R. (1985). Rethinking the sequence and priorities of high school mathematics curricula. In C. Hirsch & M. Zweng (Eds.), The secondary school mathematics curriculum (Yearbook of the National Council of Teachers of Mathematics) (pp. 43-52). Reston, Va.: National Council of Teachers of Mathematics.    Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels: building meaning for symbols and their manipulation. Second handbook of research on mathematics teaching and learning: a project of the National Council of Teachers of Mathematics (pp. 707-762). Charlotte, N.C.: Information Age Pub.    Pimm, D. (1995). Symbols and meanings in school mathematics. London and New York: Routledge.    Weems, R. (2007). Teaching math as a language. Indianapolis, Ind.: Dog Ear Publishing.